Three-dimensional, volume-preserving maps and flows and applications in fluid dynamics
by
Igor Mezic
Dept. of Engineering and Applied Science, Harvard University, Cambridge, Massachsetts, USA

We present some developments in theory of three-dimensional maps and flows. Symmetry considerations are used to discuss their integrability and the resulting structure used to study the dynamics of maps and flows perturbatively. Two characteristic classes arise: 1) action-action-angle and 2) action-angle-angle maps and flows. Class 2) has similar characteristics to two-dimensional symplectic systems while class 1) is quite different. For example, an analogue of the Kolmogorov-Arnold-Moser theorem on persistence of tori upon perturbation is valid in class 2). In contrast, generically any invariant surface of an integrable map of class 1) will not persist upon perturbation. Applications to the theory of mixing in incompressible fluid flows are discussed.

Date received: May 17, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-86.